superalgebra

and

supergeometry

# Contents

## Idea

Where a system of quantum mechanics is specified by

a system of supersymmetric quantum mechanics has

• a super Hilbert space $ℋ$;

• an odd operator $D$ in $ℋ$, the supercharge

• such that $D\circ D=H$ is the Hamiltonian.

If we regard the Hamiltonian as the generator of the Poincare Lie algebra in one dimension, then the graded commutator $\left[D,D\right]=2H$ is the supersymmetry extension to the super Poincare Lie algebra.

The data of a system of supersymmetric quantum mechanics may also be formalized in terms of a spectral triple.

## References

• Edward Witten, Supersymmetry and Morse theory J. Differential Geom. Volume 17, Number 4 (1982), 661-692. (Euclid)
• Fred Cooper, Avinash Khare, Uday Sukhatme, Supersymmetry and Quantum Mechanics Physics Reports Volume 251 (1995), 267-385. arXiv:hep-th/9405029
Revised on May 17, 2013 03:13:05 by Urs Schreiber (82.169.65.155)